MAT3701 Assignment 2 2026 | Due 5 June 2026 - Distinction Guaranteed

MAT3701
ASSIGNMENT 2 2026
DUE: 4 JUNE 2026 (MEMO)

, MAT3701 ASSIGNMENT 2 2026
DUE 5 JUNE 2026




Question 1 (20 marks)

(1.1) Consider the transformation
T : P2 (R) → P3 (R)
f (x) 7→ xf (x) + f ′ (x).

Question 1

(1.1.1) Prove that T is linear.
Let f , g ∈ P2 (R) and c ∈ R.
​




T (f + g) = x(f + g)(x) + (f + g)′ (x)
= x[f (x) + g(x)] + [f ′ (x) + g ′ (x)]
​
= xf (x) + xg(x) + f ′ (x) + g ′ (x) ​




= [xf (x) + f ′ (x)] + [xg(x) + g ′ (x)]
= T (f ) + T (g).
​ ​




T (cf ) = x(cf (x)) + (cf )′ (x)
= cxf (x) + cf ′ (x)
= c[xf (x) + f ′ (x)] = cT (f ).

T satisfies both linearity conditions.
(Reference: Friedberg §2.1, Definition of linear transformation)




(1.1.2) Basis for N (T ) [2]

Let f (x) = ax2 + bx + c. Then f ′ (x) = 2ax + b.

T (f ) = x(ax2 + bx + c) + (2ax + b) = ax3 + bx2 + (c + 2a)x + b.

Setting T (f )
= 0 (zero polynomial) gives:
a = 0, b = 0, c + 2a = 0 ⇒ c = 0. Hence f ≡ 0.
Thus N (T ) = {0}, basis = ∅ (dimension 0).